I realize this question isn't strictly mathematical, and if it doesn't fit with the content on this site then feel free (moderators/high-rep users) to close it. But when I thought up the question it seemed to me that the users on this site would be best equipped to answer it.

I've always been intrigued by the paintings of M.C. Escher comprising infinitely repeating patterns. For those of you not familiar with his work, here's an example, titled Angels & Devils:

Maybe if I were to attempt such a painting myself, today, I could find a computer program to generate random fractal-like patterns over and over until I found one I felt I could work with; then I could simply "fill in the space" with whatever image I chose. But surely Escher didn't have any such tool available to him, right?.

So: how might Escher have designed such patterns? Does anyone have any mathematical insight into what process might have been used to accomplish this? Alternatively, does anyone possibly have some historical knowledge of how Escher actually did do this?

6 Answers
6

'Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's works Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, "Escher got it absolutely right to the millimeter."'

The June/July 2010 issue of the AMS Notices here has a further article by Doris Schattschneider (a graduate school classmate of mine) on Coxeter and Escher. Doris has written extensively about Escher's work from a mathematical viewpoint, so it's worth checking out her point of view.

P.S. Don't overlook in this Notices issue Bill Casselman's column About the
Cover.

When I was in high-school, my teacher suggested a very interesting method for generating some interesting Escher's like pictures. I think it is useful for you.

Get a mirror(or something like mirror) and construct some geometrical shape with this mirror, for example an cylinder, that the inner surface of this shape is mirror. Now, collect some isomorphic shape, for example shell, flower, or simply some triangles.
If we put these objects inside the cylinder, we find very beautiful Escher-like shape on the mirror.
With some simple method in photography, we can obtain the picture that is inside the cylinder.
We can change the shape of mirror and create some elegant symmetry.

Around 1976, I became enthralled by Eschers patterns. I purchased many books on his art, hoping to find his method, but as far as I could find, there weren't any published. I eventually developed a system on my own by which I could create patterns at will. I have since created about thirty different patterns of my own which include elephants, eagles, angels, bisons, marine reptiles, leaves, gargoyles, dragons, etc. This system I developed, I believe, is the same, or very similar to what Escher employed, because I have since seen books that show the underlying gridwork, and graduation from the basic square grid, to the final curves of the finished pattern. This obviously doesn't require a computer, just some graph paper, a pencil with eraser, and some basic understanding of patterns along with a minor amount of artistic creativity. The graph paper can be a basic square grid, or even triangular or hexagonal, or any other regular gridwork. In a nutshell, simplify your animal or plant shape to its very basic shape, using the grid to block it out, and using common dimensions as much as possible, make a pattern of that block shape. By understanding rotation point s-curves,etc, the block can be gradually transformed into curves. The entire system is too lengthy to go into here, but you may see some of my results on my Alfredman/facebook page which is open to the public. The page is Identified as comic strip and creative writings. I have included some of my patterns on that page,and if any interest is shown, I could go into greater detail on the subject.As far as the circular design works, a grid can be formed using a compass and a straightedge, and a previously designed pattern can be applied to the resulting grid. A basic example of this is also included on my page.

In addition, as to the question of whether this fits as a mathematical query, it does. Patterns are a form of math. Not all math directly concerns numerical computations, geometry is math, as is music. Math can be intuitive, and creative. Check out Eschers book called Fantasy and Symmentry, by Caroline H. Macgillavry, Abrams press. This book contains the periodic drawings of M C Escher, and is used to illustrate how crystals form, basically a crystalography textbook. It explains some various elements of pattern formation, translation, rotation, mirroring etc. In MY patterns, as in music, a form of Jazz can be incorporated, by including a bit of randomizing. By this I mean that while the shapes have a definite structure and regularity, The colors do not have to conform to a regular pattern, although they certainly can. Escher also does this by way of morphing his shapes within a pattern to cause change. Much like visual music.

I have to laugh, I asked a question of my own today, framed it as well as I could, gained a student badge for getting one up vote, but was then closed down for it not being deemed a question, and lost twenty reputation points. wow
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alfred myersJan 6 '12 at 1:41